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Theorem csbima12 5344
Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Revised by NM, 20-Aug-2018.)
Assertion
Ref Expression
csbima12  |-  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B )

Proof of Theorem csbima12
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3423 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ ( F
" B )  = 
[_ A  /  x ]_ ( F " B
) )
2 csbeq1 3423 . . . . 5  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3423 . . . . 5  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
42, 3imaeq12d 5328 . . . 4  |-  ( y  =  A  ->  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
51, 4eqeq12d 2465 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ ( F " B
)  =  ( [_ y  /  x ]_ F "
[_ y  /  x ]_ B )  <->  [_ A  /  x ]_ ( F " B )  =  (
[_ A  /  x ]_ F " [_ A  /  x ]_ B ) ) )
6 vex 3098 . . . 4  |-  y  e. 
_V
7 nfcsb1v 3436 . . . . 5  |-  F/_ x [_ y  /  x ]_ F
8 nfcsb1v 3436 . . . . 5  |-  F/_ x [_ y  /  x ]_ B
97, 8nfima 5335 . . . 4  |-  F/_ x
( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
10 csbeq1a 3429 . . . . 5  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
11 csbeq1a 3429 . . . . 5  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
1210, 11imaeq12d 5328 . . . 4  |-  ( x  =  y  ->  ( F " B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B ) )
136, 9, 12csbief 3445 . . 3  |-  [_ y  /  x ]_ ( F
" B )  =  ( [_ y  /  x ]_ F " [_ y  /  x ]_ B )
145, 13vtoclg 3153 . 2  |-  ( A  e.  _V  ->  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
15 csbprc 3807 . . 3  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F " B )  =  (/) )
16 csbprc 3807 . . . . 5  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ B  =  (/) )
1716imaeq2d 5327 . . . 4  |-  ( -.  A  e.  _V  ->  (
[_ A  /  x ]_ F " [_ A  /  x ]_ B )  =  ( [_ A  /  x ]_ F " (/) ) )
18 ima0 5342 . . . 4  |-  ( [_ A  /  x ]_ F "
(/) )  =  (/)
1917, 18syl6req 2501 . . 3  |-  ( -.  A  e.  _V  ->  (/)  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
2015, 19eqtrd 2484 . 2  |-  ( -.  A  e.  _V  ->  [_ A  /  x ]_ ( F " B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B ) )
2114, 20pm2.61i 164 1  |-  [_ A  /  x ]_ ( F
" B )  =  ( [_ A  /  x ]_ F " [_ A  /  x ]_ B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1383    e. wcel 1804   _Vcvv 3095   [_csb 3420   (/)c0 3770   "cima 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002
This theorem is referenced by:  csbrn  5458  disjpreima  27421
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